{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 期末作业：预测美国股票市场收益率"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 922,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np # 数据处理最重要的模块\n",
    "import pandas as pd # 数据处理最重要的模块\n",
    "import scipy.stats as stats # 统计模块\n",
    "import scipy\n",
    "# import pymysql  # 导入数据库模块\n",
    "\n",
    "from datetime import datetime # 时间模块\n",
    "import statsmodels.formula.api as smf  # OLS regression\n",
    "\n",
    "# import pyreadr # read RDS file\n",
    "\n",
    "from matplotlib import style\n",
    "import matplotlib.pyplot as plt  # 画图模块\n",
    "import matplotlib.dates as mdates\n",
    "\n",
    "from matplotlib.font_manager import FontProperties # 作图中文\n",
    "from pylab import mpl\n",
    "#mpl.rcParams['font.sans-serif'] = ['SimHei']\n",
    "#plt.rcParams['font.family'] = 'Times New Roman'\n",
    "\n",
    "from pandas.tseries.offsets import MonthEnd\n",
    "\n",
    "#输出矢量图 渲染矢量图\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "from IPython.core.interactiveshell import InteractiveShell # jupyter运行输出的模块\n",
    "#显示每一个运行结果\n",
    "InteractiveShell.ast_node_interactivity = 'all'\n",
    "\n",
    "#设置行不限制数量\n",
    "#pd.set_option('display.max_rows',None)\n",
    "\n",
    "#设置列不限制数量\n",
    "pd.set_option('display.max_columns', None)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.导入数据，数据预处理"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 923,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>r</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>month</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1926-01-01</th>\n",
       "      <td>0.000561</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-02-01</th>\n",
       "      <td>-0.033046</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-03-01</th>\n",
       "      <td>-0.064002</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-04-01</th>\n",
       "      <td>0.037029</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-05-01</th>\n",
       "      <td>0.012095</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023-08-01</th>\n",
       "      <td>-0.020300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023-09-01</th>\n",
       "      <td>-0.048003</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023-10-01</th>\n",
       "      <td>-0.029225</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023-11-01</th>\n",
       "      <td>0.093358</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023-12-01</th>\n",
       "      <td>0.053768</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1176 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                   r\n",
       "month               \n",
       "1926-01-01  0.000561\n",
       "1926-02-01 -0.033046\n",
       "1926-03-01 -0.064002\n",
       "1926-04-01  0.037029\n",
       "1926-05-01  0.012095\n",
       "...              ...\n",
       "2023-08-01 -0.020300\n",
       "2023-09-01 -0.048003\n",
       "2023-10-01 -0.029225\n",
       "2023-11-01  0.093358\n",
       "2023-12-01  0.053768\n",
       "\n",
       "[1176 rows x 1 columns]"
      ]
     },
     "execution_count": 923,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "US_market_ret = pd.read_csv('C:/Users/21230/Desktop/Python-homework/python-homework-ykl-1/US stock market return.csv')\n",
    "US_market_ret['month'] = pd.to_datetime(US_market_ret['month'],format='%Y-%m-%d')\n",
    "US_market_ret.set_index('month',inplace=True)\n",
    "US_market_ret.sort_index(inplace=True)\n",
    "US_market_ret"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.描述性统计，统计检验"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 描述性统计"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 基础描述"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 924,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "count    1176.000000\n",
       "mean        0.009298\n",
       "std         0.053350\n",
       "min        -0.291731\n",
       "25%        -0.018340\n",
       "50%         0.012866\n",
       "75%         0.039376\n",
       "max         0.394143\n",
       "Name: r, dtype: float64"
      ]
     },
     "execution_count": 924,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "US_market_ret['r'].describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "均值是0.009298 \n",
    "\n",
    "标准差是0.053350\n",
    "- 第一组 -0.291731<=R<=-0.018340\n",
    "- 第二组 -0.018340<R<=0.012866\n",
    "- 第三组 0.012866<R<=0.039376\n",
    "- 第四组 0.039376<R<=0.394143"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 均值"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\bar{R}=\\frac{\\sum(R)}{N}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 925,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.009298055272108844"
      ]
     },
     "execution_count": 925,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(US_market_ret['1926':'2023']['r'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 926,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.009298"
      ]
     },
     "execution_count": 926,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "US_market_ret['1926':'2023']['r'].mean().round(6)#保留六位小数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 927,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0093"
      ]
     },
     "execution_count": 927,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round(US_market_ret['1926':'2023']['r'].mean(),4)#保留四位小数的另外一种写法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 928,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "美国股票市场平均月收益率为:0.9298%\n"
     ]
    }
   ],
   "source": [
    "print('美国股票市场平均月收益率为:',round(US_market_ret['1926':'2023']['r'].mean()*100,4),\"%\",sep='')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 929,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "美国股票市场平均季度收益率为:2.9418%\n"
     ]
    }
   ],
   "source": [
    "Quarter_data = US_market_ret.resample('Q')['r'].apply(lambda x:(1+x).prod()-1).to_frame()\n",
    "print('美国股票市场平均季度收益率为:',round(Quarter_data['1926':'2023']['r'].mean()*100,4),\"%\",sep='')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 930,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.009298055272108861"
      ]
     },
     "execution_count": 930,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(US_market_ret['1926':'2023']['r']) / len(US_market_ret['1926':'2023']['r'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 偏度 skewness"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "s=\\frac{1}{N} \\sum^{N}_{i=1} [(\\frac{R-\\bar{R}}{\\sigma}^3)]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 931,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6275510204081632"
      ]
     },
     "execution_count": 931,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(US_market_ret[US_market_ret['r'] > 0]) / len(US_market_ret)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 统计检验"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 因为n>30，所以选择z检验比较好"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ Z = \\frac{\\bar{X} - \\mu}{\\sigma/ \\sqrt{n}} $$\n",
    "其中,xbar为样本均值, σ为总体标准差, n为样本容量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 932,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 1000x600 with 0 Axes>"
      ]
     },
     "execution_count": 932,
     "metadata": {},
     "output_type": "execute_result"
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    {
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     "execution_count": 932,
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   "source": [
    "# 设置参数\n",
    "alpha = 0.05  # 显著性水平\n",
    "z_crit = stats.norm.ppf(1 - alpha / 2)  # 双侧检验临界值\n",
    "\n",
    "# 生成 z 分布的 x 轴数据\n",
    "x = np.linspace(-4, 4, 1000)  # 生成 -4 到 4 之间的 1000 个数据\n",
    "y = stats.norm.pdf(x)  # 计算标准正态分布的概率密度函数\n",
    "\n",
    "# 绘制 z 分布曲线\n",
    "plt.figure(figsize=(10, 6))  # 设置图形大小\n",
    "plt.style.use('ggplot')  # 设置风格\n",
    "plt.plot(x, y, label='Standard Normal Distribution', color='blue')  # 绘制 z 分布曲线\n",
    "\n",
    "# 填充拒绝原假设区域\n",
    "plt.fill_between(x, 0, y, where=(x >= z_crit) | (x <= -z_crit), color='red', alpha=0.7, label='Rejection region')\n",
    "\n",
    "# 标出 z 临界值\n",
    "plt.axvline(z_crit, color='red', linestyle='--', label=f'Critical value = {z_crit:.2f}')  # 标出临界值\n",
    "plt.axvline(-z_crit, color='red', linestyle='--')  # 标出临界值\n",
    "\n",
    "# 假设计算出的 z 值 \n",
    "z_value = (sample_mean - population_mean) / (sample_std / np.sqrt(sample_size))\n",
    "plt.axvline(z_value, color='green', linestyle='-', label=f'z-value = {z_value}')\n",
    "\n",
    "# 图形标注\n",
    "plt.title('Standard Normal Distribution with Rejection Regions and z-value')\n",
    "plt.xlabel('z-value')\n",
    "plt.ylabel('Probability Density')\n",
    "plt.legend()\n",
    "\n",
    "# 显示图形\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### t检验"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ t = \\frac{\\bar{X} - \\mu}{\\sigma/ \\sqrt{n}} $$\n",
    "其中,xbar为样本均值, σ为总体标准差, n为样本容量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 933,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TtestResult(statistic=array([5.92162899]), pvalue=array([4.18300932e-09]), df=array([1172]))"
      ]
     },
     "execution_count": 933,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.ttest_1samp(US_market_ret['1926-01':'2023-09'], 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.图表展示"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 美国股票市场收益率的直方图"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 934,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 1000x600 with 0 Axes>"
      ]
     },
     "execution_count": 934,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "        161., 152., 115.,  89.,  51.,  21.,   9.,   8.,   8.,   3.,   0.,\n",
       "          2.,   0.,   0.,   1.,   0.,   1.,   0.,   0.,   0.,   0.,   0.,\n",
       "          0.,   0.,   1.,   0.,   1.,   1.]),\n",
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       "         0.394143  ]),\n",
       " <BarContainer object of 50 artists>)"
      ]
     },
     "execution_count": 934,
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      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "returns = US_market_ret['r']\n",
    "# 绘制直方图\n",
    "plt.figure(figsize=(10, 6)) \n",
    "plt.hist(returns, bins=50, color='#00F5FF', alpha=0.7, edgecolor='black') \n",
    "plt.title('Histogram of U.S. Stock Market Returns')  \n",
    "plt.xlabel('Return') \n",
    "plt.ylabel('Frequency') \n",
    "plt.grid(True) \n",
    "plt.show()  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 美国股票市场收益率的时序折线图"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 935,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 1500x600 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(15, 6)) # 图片比例\n",
    "ax.plot(\n",
    "    'r',  # 要画图的变量名\n",
    "    '.-',  # 线的类型\n",
    "    color = '#FF6A6A', \n",
    "    label = 'Return',  \n",
    "    linewidth = 1.2,  \n",
    "    data = US_market_ret['1926':'2023'] )\n",
    "ax.set_title(\"Time Series of U.S. Stock Market Returns\") \n",
    "ax.set_xlabel('month') \n",
    "plt.ylabel('Return') \n",
    "\n",
    "# 设置x轴的日期格式\n",
    "date_format = mdates.DateFormatter('%Y')\n",
    "ax.xaxis.set_major_formatter(date_format)\n",
    "\n",
    "# # 设置x轴的刻度间隔\n",
    "ax.xaxis.set_major_locator(mdates.YearLocator(base=5))\n",
    "\n",
    "# 旋转x轴标签以防止重叠\n",
    "plt.xticks(rotation=90)\n",
    "\n",
    "# 添加图例\n",
    "plt.legend(loc='upper left', frameon=False, fontsize=10)\n",
    "fig.savefig('Monthly_return.pdf', bbox_inches='tight')# 更改输出图片格式 jpg\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.可预测性 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 选用的CPI数据的描述性统计、统计检验、图表展示"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 导入数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 936,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>cpi</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>month</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1970-01-01</th>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-02-01</th>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-03-01</th>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-04-01</th>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-05-01</th>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2024-08-14</th>\n",
       "      <td>-0.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2024-09-11</th>\n",
       "      <td>0.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2024-10-10</th>\n",
       "      <td>0.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2024-11-13</th>\n",
       "      <td>0.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2024-12-11</th>\n",
       "      <td>0.2</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>660 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            cpi\n",
       "month          \n",
       "1970-01-01  NaN\n",
       "1970-02-01  0.5\n",
       "1970-03-01  0.5\n",
       "1970-04-01  0.5\n",
       "1970-05-01  0.5\n",
       "...         ...\n",
       "2024-08-14 -0.1\n",
       "2024-09-11  0.2\n",
       "2024-10-10  0.2\n",
       "2024-11-13  0.2\n",
       "2024-12-11  0.2\n",
       "\n",
       "[660 rows x 1 columns]"
      ]
     },
     "execution_count": 936,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cpi=pd.read_csv('C:/Users/21230/Desktop/Python-homework/python-homework-ykl-1/美国消费者物价指数月率CPI数据.csv')\n",
    "cpi['month'] = pd.to_datetime(cpi['month'],format='%Y/%m/%d')\n",
    "cpi.set_index('month',inplace=True)\n",
    "cpi.sort_index(inplace=True)\n",
    "cpi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 为什么选用CPI来预测美国股票市场收益率：CPI是衡量通货膨胀的重要指标，直接影响企业的成本和利润，进而影响股票收益率。当CPI上升时，企业成本增加，利润可能被压缩，同时可能导致美联储加息，增加企业融资成本和降低消费者购买力，从而对股市产生负面影响。相反，CPI下降时，企业成本压力减轻，利润空间扩大，且货币政策可能保持宽松，有利于股市上涨。因此，CPI是预测股票收益率的重要参考因素之一。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 描述性统计"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 937,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "count    659.000000\n",
       "mean       0.322762\n",
       "std        0.334761\n",
       "min       -1.700000\n",
       "25%        0.200000\n",
       "50%        0.300000\n",
       "75%        0.500000\n",
       "max        1.800000\n",
       "Name: cpi, dtype: float64"
      ]
     },
     "execution_count": 937,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#基础描述\n",
    "cpi['cpi'].describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 938,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8621212121212121"
      ]
     },
     "execution_count": 938,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#偏度\n",
    "len(cpi[cpi['cpi'] > 0]) / len(cpi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 统计检验 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ t = \\frac{\\bar{X} - \\mu}{\\sigma/ \\sqrt{n}} $$\n",
    "其中,xbar为样本均值, σ为总体标准差, n为样本容量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 939,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TtestResult(statistic=array([24.68475258]), pvalue=array([2.93469091e-85]), df=array([443]))"
      ]
     },
     "execution_count": 939,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.ttest_1samp(cpi['1971':'2007'], 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 两变量合到一起"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 940,
   "metadata": {},
   "outputs": [
    {
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       "      <th>1970-01-01</th>\n",
       "      <td>-0.073254</td>\n",
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       "      <th>1970-02-01</th>\n",
       "      <td>0.056706</td>\n",
       "      <td>0.5</td>\n",
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       "    <tr>\n",
       "      <th>1970-03-01</th>\n",
       "      <td>-0.004729</td>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-04-01</th>\n",
       "      <td>-0.105318</td>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-05-01</th>\n",
       "      <td>-0.064346</td>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "    <tr>\n",
       "      <th>2007-10-01</th>\n",
       "      <td>0.025923</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007-11-01</th>\n",
       "      <td>-0.049415</td>\n",
       "      <td>0.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007-12-01</th>\n",
       "      <td>-0.004288</td>\n",
       "      <td>0.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008-01-01</th>\n",
       "      <td>-0.062356</td>\n",
       "      <td>0.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008-02-01</th>\n",
       "      <td>-0.022002</td>\n",
       "      <td>0.3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>458 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                   r  cpi\n",
       "month                    \n",
       "1970-01-01 -0.073254  NaN\n",
       "1970-02-01  0.056706  0.5\n",
       "1970-03-01 -0.004729  0.5\n",
       "1970-04-01 -0.105318  0.5\n",
       "1970-05-01 -0.064346  0.5\n",
       "...              ...  ...\n",
       "2007-10-01  0.025923  0.0\n",
       "2007-11-01 -0.049415  0.4\n",
       "2007-12-01 -0.004288  0.3\n",
       "2008-01-01 -0.062356  0.8\n",
       "2008-02-01 -0.022002  0.3\n",
       "\n",
       "[458 rows x 2 columns]"
      ]
     },
     "execution_count": 940,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "reg_data = pd.merge(US_market_ret, cpi, left_index=True, right_index=True)\n",
    "reg_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 图表展示"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 941,
   "metadata": {},
   "outputs": [
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_11\">\n",
       "     <g id=\"line2d_21\">\n",
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       "     </g>\n",
       "     <g id=\"line2d_22\">\n",
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       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 1979 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(273.497421 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_12\">\n",
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       "     <g id=\"line2d_24\">\n",
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       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 1980 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(293.161243 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_13\">\n",
       "     <g id=\"line2d_25\">\n",
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       "     </g>\n",
       "     <g id=\"line2d_26\">\n",
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       "      </g>\n",
       "     </g>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_14\">\n",
       "     <g id=\"line2d_27\">\n",
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       "     <g id=\"line2d_28\">\n",
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       "      </g>\n",
       "     </g>\n",
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       "      <g style=\"fill: #555555\" transform=\"translate(332.54276 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_15\">\n",
       "     <g id=\"line2d_29\">\n",
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       "     </g>\n",
       "     <g id=\"line2d_30\">\n",
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       "       <use xlink:href=\"#m03c6a5d91a\" x=\"349.447206\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_16\">\n",
       "     <g id=\"line2d_31\">\n",
       "      <path d=\"M 369.111028 356.78175 \n",
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       "     </g>\n",
       "     <g id=\"line2d_32\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"369.111028\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_16\">\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_17\">\n",
       "     <g id=\"line2d_33\">\n",
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       "     </g>\n",
       "     <g id=\"line2d_34\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"388.828723\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_17\">\n",
       "      <!-- 1985 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(391.588098 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_18\">\n",
       "     <g id=\"line2d_35\">\n",
       "      <path d=\"M 408.492545 356.78175 \n",
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       "\" clip-path=\"url(#p1a6d932cb0)\" style=\"fill: none; stroke: #ffffff; stroke-width: 0.8; stroke-linecap: square\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_36\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"408.492545\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_18\">\n",
       "      <!-- 1986 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(411.25192 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"127.246094\"/>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_19\">\n",
       "     <g id=\"line2d_37\">\n",
       "      <path d=\"M 428.156367 356.78175 \n",
       "L 428.156367 24.14175 \n",
       "\" clip-path=\"url(#p1a6d932cb0)\" style=\"fill: none; stroke: #ffffff; stroke-width: 0.8; stroke-linecap: square\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_38\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"428.156367\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_19\">\n",
       "      <!-- 1987 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(430.915742 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-37\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_20\">\n",
       "     <g id=\"line2d_39\">\n",
       "      <path d=\"M 447.820189 356.78175 \n",
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       "\" clip-path=\"url(#p1a6d932cb0)\" style=\"fill: none; stroke: #ffffff; stroke-width: 0.8; stroke-linecap: square\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_40\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"447.820189\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_20\">\n",
       "      <!-- 1988 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(450.579564 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_21\">\n",
       "     <g id=\"line2d_41\">\n",
       "      <path d=\"M 467.537884 356.78175 \n",
       "L 467.537884 24.14175 \n",
       "\" clip-path=\"url(#p1a6d932cb0)\" style=\"fill: none; stroke: #ffffff; stroke-width: 0.8; stroke-linecap: square\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_42\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"467.537884\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_21\">\n",
       "      <!-- 1989 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(470.297259 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_22\">\n",
       "     <g id=\"line2d_43\">\n",
       "      <path d=\"M 487.201706 356.78175 \n",
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       "     </g>\n",
       "     <g id=\"line2d_44\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"487.201706\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_22\">\n",
       "      <!-- 1990 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(489.961081 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_23\">\n",
       "     <g id=\"line2d_45\">\n",
       "      <path d=\"M 506.865527 356.78175 \n",
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       "     </g>\n",
       "     <g id=\"line2d_46\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"506.865527\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
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       "      <g style=\"fill: #555555\" transform=\"translate(509.624902 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" x=\"190.869141\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_24\">\n",
       "     <g id=\"line2d_47\">\n",
       "      <path d=\"M 526.529349 356.78175 \n",
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       "     </g>\n",
       "     <g id=\"line2d_48\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"526.529349\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_24\">\n",
       "      <!-- 1992 -->\n",
       "      <g style=\"fill: #555555\" transform=\"translate(529.288724 389.23175) rotate(-90) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"127.246094\"/>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_25\">\n",
       "     <g id=\"line2d_49\">\n",
       "      <path d=\"M 546.247044 356.78175 \n",
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       "     </g>\n",
       "     <g id=\"line2d_50\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"546.247044\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
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       "     </g>\n",
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       "      <!-- 1993 -->\n",
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       "       <use xlink:href=\"#DejaVuSans-39\" x=\"63.623047\"/>\n",
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       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_26\">\n",
       "     <g id=\"line2d_51\">\n",
       "      <path d=\"M 565.910866 356.78175 \n",
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       "     </g>\n",
       "     <g id=\"line2d_52\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"565.910866\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
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       "      <!-- 1994 -->\n",
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       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_27\">\n",
       "     <g id=\"line2d_53\">\n",
       "      <path d=\"M 585.574688 356.78175 \n",
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       "     <g id=\"line2d_54\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m03c6a5d91a\" x=\"585.574688\" y=\"356.78175\" style=\"fill: #555555; stroke: #555555; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_27\">\n",
       "      <!-- 1995 -->\n",
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       "    </g>\n",
       "    <g id=\"xtick_33\">\n",
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       "    </g>\n",
       "    <g id=\"xtick_34\">\n",
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       "    <g id=\"xtick_35\">\n",
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       "    </g>\n",
       "    <g id=\"xtick_36\">\n",
       "     <g id=\"line2d_71\">\n",
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       "    <g id=\"xtick_37\">\n",
       "     <g id=\"line2d_73\">\n",
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       "    <g id=\"xtick_38\">\n",
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       "    <g id=\"xtick_39\">\n",
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      "text/plain": [
       "<Figure size 1500x600 with 2 Axes>"
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     "output_type": "display_data"
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   "source": [
    "fig, ax1 = plt.subplots(figsize=(15,6))\n",
    "# the linewidth and marker size are set to be very small\n",
    "ax1.plot(US_market_ret['r']['1971':'2008'],color='#008B8B',marker='*',linewidth=0.9,\n",
    "         markersize=4,\n",
    "         linestyle='--',label='US Stock Market Return')\n",
    "ax1.set_ylabel('US Stock Market Return',color='c')\n",
    "#ax1.set_xlabel('Month')\n",
    "\n",
    "# 设置x轴的日期显示格式\n",
    "data_format = mdates.DateFormatter('%Y')\n",
    "ax1.xaxis.set_major_formatter(data_format)\n",
    "ax1.xaxis.set_major_locator(mdates.YearLocator())\n",
    "\n",
    "# 转置x轴的日期显示格式\n",
    "plt.xticks(rotation = 90)\n",
    "\n",
    "ax2 = ax1.twinx()\n",
    "ax2.plot(reg_data['cpi'].shift(2),color='blue',marker='.',linewidth=0.9,\n",
    "         markersize=4,\n",
    "         linestyle='--',label='cpi')\n",
    "\n",
    "ax2.set_ylabel('cpi',color='r')\n",
    "\n",
    "plt.title('US Stock Market Return and CPI')\n",
    "\n",
    "# change the legend into one box\n",
    "lines, labels = ax1.get_legend_handles_labels()\n",
    "lines2, labels2 = ax2.get_legend_handles_labels()\n",
    "ax2.legend(lines + lines2, labels + labels2, loc='upper left')\n",
    "\n",
    "ax.xaxis.set_major_locator(mdates.YearLocator(base=5))\n",
    "\n",
    "# save figure\n",
    "\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 美国股票市场收益率进行预测"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$r_{t} = \\alpha + \\beta x_{t-1} + \\epsilon_{t}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 942,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      r   R-squared:                       0.009\n",
      "Model:                            OLS   Adj. R-squared:                 -0.008\n",
      "Method:                 Least Squares   F-statistic:                     2.149\n",
      "Date:                Sat, 04 Jan 2025   Prob (F-statistic):              0.148\n",
      "Time:                        19:47:29   Log-Likelihood:                 131.01\n",
      "No. Observations:                  59   AIC:                            -258.0\n",
      "Df Residuals:                      57   BIC:                            -253.9\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:                  HAC                                         \n",
      "==============================================================================\n",
      "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "Intercept      0.0107      0.003      3.227      0.001       0.004       0.017\n",
      "cpi            0.8539      0.582      1.466      0.143      -0.288       1.995\n",
      "==============================================================================\n",
      "Omnibus:                        0.175   Durbin-Watson:                   1.815\n",
      "Prob(Omnibus):                  0.916   Jarque-Bera (JB):                0.027\n",
      "Skew:                          -0.052   Prob(JB):                        0.986\n",
      "Kurtosis:                       2.989   Cond. No.                         333.\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors are heteroscedasticity and autocorrelation robust (HAC) using 6 lags and without small sample correction\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\21230\\AppData\\Local\\Temp\\ipykernel_9160\\50082808.py:2: SettingWithCopyWarning: \n",
      "A value is trying to be set on a copy of a slice from a DataFrame.\n",
      "Try using .loc[row_indexer,col_indexer] = value instead\n",
      "\n",
      "See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n",
      "  reg_data_1['cpi'] = reg_data_1['cpi'].shift(2)/100\n"
     ]
    }
   ],
   "source": [
    "reg_data_1=reg_data['1971':'2007']\n",
    "reg_data_1['cpi'] = reg_data_1['cpi'].shift(2)/100\n",
    "model_teim = smf.ols('r ~ cpi',\n",
    "                 data=reg_data_1['2003-02':'2023-09']).fit(\n",
    "                     cov_type='HAC', cov_kwds={'maxlags': 6})\n",
    "print(model_teim.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 理论论述和结果解释"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 模型拟合度:\n",
    "    R-squared: 0.009，表示模型只能解释因变量（r）变异的0.9%。这是一个非常低的值，说明模型的解释力很弱。\n",
    "    Adj. R-squared: 0.008，调整后的 R 平方值是负数，这进一步表明模型可能不适合数据。\n",
    "\n",
    "2. F统计量和其概率:\n",
    "    F-statistic: 2.149，这是模型整体显著性的统计量。\n",
    "    Prob (F-statistic): 0.148，这个值大于0.05，意味着我们不能拒绝零假设，即模型中的预测变量对因变量没有显著影响。\n",
    "\n",
    "3. 系数估计:\n",
    "    Intercept: 0.0107，表示当 cpi 为0时，r 的预期值。\n",
    "    cpi: 0.8539，表示 cpi 每增加一个单位，r 预期增加 0.8539 个单位，但这个系数的 P 值（0.143）大于 0.05，说明它在统计上不显著。\n",
    "\n",
    "4. 置信区间:\n",
    "    Intercept: [0.004, 0.017]，表示我们95%确信截距的真实值在这个区间内。\n",
    "    cpi: [0.288, 1.995]，cpi 的置信区间包含了0，这进一步表明它可能在统计上不显著。\n",
    "\n",
    "5. 诊断统计量:\n",
    "    Omnibus: 0.175，Prob(Omnibus): 0.916，这个测试检验残差的正态性，高 p 值表明残差可能是正态分布的。\n",
    "    DurbinWatson: 1.815，这个值接近2，表明残差之间没有自相关。\n",
    "    JarqueBera(JB): 0.027，Prob(JB): 0.986，这个测试也检验残差的正态性，高 p 值同样表明残差可能是正态分布的。\n",
    "    Skew: 0.052，表明残差分布的偏度接近对称。\n",
    "    Kurtosis: 2.989，接近正态分布的峰度（3）。\n",
    "    Cond. No.: 333.，条件数相对较高，可能表明存在多重共线性问题。\n",
    "\n",
    "综上，该模型的拟合度很低，预测变量 cpi 对因变量 r 的影响在统计上不显著。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为什么用美国CPI在OSL算法上预测美国股票市场收益率效果并不好：\n",
    "\n",
    "1. CPI作为单一指标的局限性：CPI主要反映的是消费者购买商品和服务的价格变动情况，它虽然能够提供关于通货膨胀水平的信息，但并不能全面反映影响股票市场收益率的所有因素。股票市场受到公司盈利、利率、市场情绪、政策变化等多种因素的影响。在某些情况下，CPI的变动可能会对股市产生影响，但这种影响可能是间接的，并且可能会被其他因素所抵消。\n",
    "\n",
    "2. 市场预期和适应性：股市往往会对已知的经济数据做出快速反应，包括CPI。一旦CPI数据公布，市场可能已经对其潜在影响进行了定价，这使得使用CPI来预测未来收益率变得更加困难。CPI作为一个经济指标，它反映的是过去的价格变动情况，而股市投资者更关注的是未来经济增长和企业盈利的预期。因此，CPI的滞后性可能限制了其作为预测工具的有效性。\n",
    "\n",
    "3. 货币政策、季节性和极端事件的影响：CPI是美联储等中央银行制定货币政策的重要参考。当CPI上升时，央行可能会提高利率以抑制通货膨胀，这会增加企业的融资成本，影响其盈利能力，进而影响股市。然而，这种影响是多方面的，并且可能会被其他经济因素所抵消。同时CPI数据可能会受到季节性因素和极端事件的影响，这些因素可能会扭曲CPI与股市之间的关系，使得基于CPI的预测变得不准确。\n",
    "\n",
    "4. OLS模型的假设、模型的过度简化、数据的非线性特征：OLS模型假设误差项是独立同分布的，且没有自相关和异方差性。如果这些假设不成立，OLS估计可能会有偏误，从而影响预测效果。使用单一变量如CPI来预测股市收益率可能过度简化了股市的复杂性。在现实世界中，股市收益率可能受到多种因素的共同作用，需要更复杂的模型来捕捉这些关系。股市收益率可能与CPI之间存在非线性关系，而OLS模型假设的是线性关系。如果实际关系是非线性的，OLS模型可能无法准确捕捉这种关系。"
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